HHT於桁架非線性動力分析之應用

Abstract

傅立葉轉換(Fourier Transformation)之應用有諸多限制條件，無法完全適用於非線性系統。為了能分析非穩態(non-stationary)或非線性過程(nonlinear processes)之訊號，本文採用具有較高適用性之希爾伯特–黃轉換(Hilbert-Huang Transform–HHT)作為分析工具之一。 希爾伯特–黃轉換主要包含兩部分之處裡流程：(1)經驗模態分離法(Empirical Mode Decomposition–EMD)：可將訊號分離成數個內建模態函數(Intrinsic Mode Function–IMF)，而每個IMF皆具有良好的希爾柏特轉換特性。(2)希爾柏特轉換(Hilbert Transform)：可得到訊號之即時頻率與即時振幅，若繪製成能量–頻率–時間分佈圖，則稱為希爾伯特頻譜(Hilbert Spectrum)。 本文以雙桿桁架系統作為測試模型，考慮幾何非線性效應，利用有限元素法配合Newmark法進行數值分析，並提供對應之達芬方程式參數。本文嘗試由頻率的角度，研究非線性系統之動力行為，比較FFT與HHT分析結果之差別，並針對振動頻率的變化、週期倍增以及混沌現象等，進行較為系統的探討。

In the Fourier analysis, the fundamental assumption of linear and stationary process is required for the data. Applying the Fourier analysis to those data generated from nonlinear systems may cause misunderstanding of the physical phenomena hidden in the data. On the other hand, the Hilbert-Huang transform (HHT) is considered more suitable for analyzing nonlinear and non-stationary data. HHT includes two major parts: (1) empirical mode decomposition (EMD): a sifting process by which the data can be decomposed into a collection of intrinsic mode functions (IMF) that admit well-behaved Hilbert transforms; (2) Hilbert transform: a type of transform by which the instantaneous frequency and amplitude can be calculated for any instant. The energy distribution being plotted in a 3-D energy-frequency-time space is designated as the Hilbert spectrum. A two-member truss system with the effect of geometric nonlinearity considered is taken as the example in this study. The dynamic response of such a system is numerically analyzed by the finite element method along with the Newmark method, with the corresponding parameters in the Duffing equation given in each case. By comparing the results obtained from both the FFT and HHT analyses in frequency domain, the dynamic behavior of the nonlinear system is systematically studied, especially with respect to the variation in frequency caused by the geometric nonlinearity, period-doubling, chaos phenomenon, and so on.